Extremal Rearrangement Problems Involving Poisson’s Equation with Robin Boundary Conditions

In this paper, we study both minimization and maximization problems corresponding to a Poisson’s equation with Robin boundary conditions. These rearrangement shape optimization problems arise in many applications including the design of mechanical vibration and fluid mechanics that explore the possibility to control the total displacement and the kinetic energy, respectively. Analytically, we study the properties of the extremizers on general domains including topology and geometry of the optimizers. Asymptotic behaviors of the optimal values are investigated as well. Although the explicit solutions are rare for this kind of optimization problems, we obtain such solutions on N-balls. Numerically, we propose efficient algorithms based on finite element methods, rearrangement techniques and our analytical results to determine the extremizers in just a few iterations on general domains.